Covering one point process with another

TL;DR

The paper analyzes the limiting behavior of the two-sample $k$-coverage threshold $R_{n,m,k}$ when $X_i$ are uniform in a region $A$ and $Y_j$ are uniform in $B$, with $m(n)\sim au n$. It combines Poissonization via Chen–Stein with careful boundary-layer analysis to derive Gumbel-type limit laws (and a two-component extreme-value law in 2D for $k=2$) that depend on dimension and boundary geometry, including explicit centering that involves $ log n$, $ log log n$, and the perimeter of $A$. The main contributions are explicit limiting distributions for $R_{n,m,k}$ under various regimes (with both smooth and polygonal 2D boundaries and higher dimensions), and a comprehensive framework connecting interior and boundary contributions through the moat region. The results have practical relevance for wireless-coverage-type problems and random geometric graphs, offering accurate asymptotics and finite-sample corrections validated by simulations. All results are established with a unified approach based on vacancy probability, Poisson approximation, and meticulous boundary-integral analysis.$

Abstract

Let $X_1,X_2, \ldots $ and $Y_1, Y_2, \ldots$ be i.i.d. random uniform points in a bounded domain $A \subset \mathbb{R}^2$ with smooth or polygonal boundary. Given $n,m,k \in \mathbb{N}$, define the {\em two-sample $k$-coverage threshold} $R_{n,m,k}$ to be the smallest $r$ such that each point of $ \{Y_1,\ldots,Y_m\}$ is covered at least $k$ times by the disks of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_{n,m,k}$ as $n \to \infty$ with $m= m(n) \sim τn$ for some constant $τ>0$, with $k $ fixed. If $A$ has unit area, then $n πR_{n,m(n),1}^2 - \log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\log τ$. For $k >2$, we find that $n πR_{n,m(n),k}^2 - \log n - (2k-3) \log \log n$ is asymptotically Gumbel with scale parameter $2$ and a more complicated location parameter involving the perimeter of $A$; boundary effects dominate when $k >2$. For $k=2$ the limiting cdf is a two-component extreme value distribution with scale parameters 1 and 2. We also give analogous results for higher dimensions, where the boundary effects dominate for all $k$.

Figures (6)

• Figure 1: Illustration for proof of Lemma \ref{['l:bdyvol']}. The set $(B_r(x) \cap A) \triangle (B_r(x) \cap \mathbb{H})$ is contained in the shaded region.
• Figure 2: Illustration for proof of Lemma \ref{['lemgeom1a']}. The segment $H'$ is centred on $x$.
• Figure 3: Illustration of the mapping $g$ in the proof of Proposition \ref{['thm:cov']}.
• Figure 4: Illustration showing the rectangles $\mathsf{Rec}_{t,i}$ (shaded) in the proof of Proposition \ref{['p:average2d']}, Case 2.
• Figure 5: The empirical distributions of $n \theta_d f_0 R_{n,m(n),k}^d - c_1 \log n - c_2 \log\log n$ obtained from computer simulations in the settings of Theorems \ref{['Hallthm']}, \ref{['thsmoothgen']} and \ref{['t:dhi']}, plotted on the same axes as the limiting distributions. See Section \ref{['sim-section']} for discussion of the simulation results.

Theorems & Definitions (31)

• Theorem 2.1: Fluctuations of $R_{n,m,k}(B)$ when $\overline{B} \subset A^o$
• Theorem 2.2: Fluctuations of $R_{n,m}$ in a planar region with boundary
• Theorem 2.3
• Definition 3.1: Sphere condition
• Lemma 3.2: Sphere condition lemma
• proof
• Remark 3.3
• Lemma 3.4
• proof
• Lemma 3.5